1. X-ray Variability of AGN
Brandon C. Kelly, Małgorzata Sobolewska, Aneta Siemiginowska
ApJ, 2011, 730, 52
5/31/11
Astrostatistics Group,

2. Quasar (Active Galactic Nuclei)
X-ray Emission
5/31/11
Astrostatistics Group,

3. AGN X-ray Variability is Aperiodic
XMM Lightcurve for MRK 766
Vaughan & Fabian (2003)
5/31/11
Astrostatistics Group,

4. What do the random fluctuations tell us?
Characteristic time scales of the fluctuations correspond to different physical mechanisms
Fluctuations may probe how the accretion flow `responds’ to a perturbation
Unable to do controlled perturbations, but turbulence (e.g., MHD effects) provides a constant source of chaotic perturbations
May be the only observational way to probe viscosity
Provides a test of GBH/SMBH connection
5/31/11
Astrostatistics Group,

5. Astrostatistics Group, bckelly@cfa.harvard.edu
Accretion flow solutions expect simple scaling of time scale with mass and accretion rate
AGN
GBHs
Predicted time scale vs. observed for GBHs and SMBHs (McHardy et al. 2006)
5/31/11
Astrostatistics Group,

6. Some Example X-ray PSDs of AGN
Observed PSDs are very information
poor, need a better statistical
technique!
Updated AKN 564 PSD
Markowitz et al. 2003, ApJ, 593, 96
McHardy et al., MNRAS, 2007, 382, 985
5/31/11
Astrostatistics Group,

7. Inadequacy of Common Methods
Time Series simulated from an
Autoregressive process
Periodogram and SF provide
poor info on variability
5/31/11
Astrostatistics Group,

8. Astrostatistics Group, bckelly@cfa.harvard.edu
A Different Approach: Use a stochastic, generative model with the right PSD
The Ornstein-Uhlenbeck (OU, autoregressive)
Process, X(t)
Continuous form:
ω0: Characteristic angular frequency
μ: Mean of X(t)
σ: Amplitude of driving noise
dW(t): A white noise process with unit variance
Discrete form:
α=exp(-ω0)
ε1, … , εi : A series of standard Gaussian random
variables
Kelly et al. (2009, ApJ, 698, 895)
5/31/11
Astrostatistics Group,

9. The PSD of the OU process is a Lorentzian
Flat, White Noise
PSD ~ 1/ω2
Red Noise ω0
‘Characteristic’ time scale:
τ=1/ω0
Note that f = ω/2π
5/31/11
Astrostatistics Group,

10. OU Process describes well the optical lightcurves of AGN
Results from Kelly et al. (2009) confirmed by Kozlowski et al.(2010), and by MacLeod et al.(2010)
OU process has been used a model for:
Variability selection of quasars (Kozlowski et al. 2010, Butler & Bloom 2010)
Reverberation mapping (Zu et al. 2010)
Probably does not capture the flaring seen in sub-mm lightcurves of blazars (Strom et al., in prep)
Kelly et al. (2009, ApJ, 698, 895)
5/31/11
Astrostatistics Group,

11. But what about X-ray lightcurves?
Use a mixture of OU processes: ω1 ωM
For both the OU process and
mixed OU process, the likelihood
function can be derived using
standard techniques
5/31/11
Astrostatistics Group,

12. Does the Mixed OU process have any physical interpretation?
Solution to the stochastic diffusion equation in a bounded medium:
y(x,t) ~ Surface Density
L(t)
x = r1/2
See also work by Titarchuk et al. (2007)
5/31/11
Astrostatistics Group,

13. Solution of Stochastic Diffusion Equation (Chow 2007)
Denote the eigenfunctions of the diffusion operator as ek(x) and the eigenvalues as ωk
Solution has the form
Suppose we can express the spatial covariances of driving noise as
In addition, random field W(x,t) can be expressed as
{wk(t)} is a sequence of brownian motions
5/31/11
Astrostatistics Group,

14. Astrostatistics Group, bckelly@cfa.harvard.edu
Solution (Continued)
We then have the set of stochastic ODEs:
This has the solution
Solution is a mixture of OU processes
5/31/11
Astrostatistics Group,

15. Astrophysical interpretation
Characteristic frequencies are the eigenvalues of the diffusion operator
Mixing weights are a combination of the eigenfunctions of the diffusion operator and the projections of the spatial covariance matrix of W(x,t) onto the space spanned by the eigenfunctions
Drift time scale at boundary
edge
Drift time scale across characteristic
spatial scale of W(x,t)
5/31/11
Astrostatistics Group,

16. The likelihood function
Mixed OU process has the state space representation:
Can use Kalman recursions to derive likelihood function, efficiently calculate it
y(t): Observed lightcurve at time t
c: Vector of mixing weights
x(t): Vector of independent OU processes at time t
ε(t): Measurement errors
5/31/11
Astrostatistics Group,

17. Application to AGN X-ray lightcurves
Characterizes the ~ 10 local Seyfert galaxies with the best X-ray lightcurves well
5/31/11
Astrostatistics Group,

18. Estimating Characteristic Timescales, other variability parameters
Based on an MCMC sampler, available
from B. Kelly
5/31/11
Astrostatistics Group,

19. Can also get flexible estimates of PSD
MCG
AKN 564
Green: Best fit flexible PSD
Red: Best fit assuming a bending power-law
Black: Random realizations of the PSD from
its probability distribution
PSDs are more ‘wiggly’ than simple
bending power-laws, similar to GBHs
5/31/11
Astrostatistics Group,

20. Trends with black hole mass
X-ray
Optical
For optical, see Kelly et al. (2009), Collier & Peterson (2001), McHardy et al. (2007), Zhou et al. (2010),
And MacLeod et al. (2010)
5/31/11
Astrostatistics Group,

21. Astrostatistics Group, bckelly@cfa.harvard.edu
Summary
X-ray Variability of AGN is well-characterized by a mixture of Ornstein-Uhlenbeck processes
Enables fitting of power spectra without Fourier transforms
Characteristic time scale associated with high-frequency break correlates well with MBH
Rate at which variability power is injection into the lightcurve tightly anti-correlated with MBH
May provide the most precise ‘cheap’ mass estimate
5/31/11
Astrostatistics Group,

22. Directions for Future Work
Extend method to work with time series of photon counts, Poisson likelihood
Extend methodology for analyzing multivariate lightcurves, more efficient and powerful than cross-correlation functions
Add in higher order terms to the stochastic ODEs to model more complicated PSDs
5/31/11
Astrostatistics Group,